How Does the Density of Q in R Get Proven in Rudin's Analysis?

[kostas230]

I have a difficulty in understanding the proof on Rudin's book page 9 regarding the density of Q in R. Specifically, I don't understand this step:

After we prove that there exist two integers $m_{1}$, $m_{2}$ with $m_{1}>nx$ and $m_{2}>-nx$ such that:

$-m_{2}<nx<m_{1}$​

What I don't understand is how from the above get's concludes the following:

Hence there is an integer m (with $-m_{2}≤m≤m_{1}$) such that:

$m-1≤nx<m$​

 

[mfb]

If m₁-1 <= nx, then set m=m₁ and both inequalities are trivial.
If m₁-1 > nx, consider m₁-2 and so on.
As the difference between m₂ and m₁ is finite, you find m in a finite number of steps.

 

[Bacle2]

Another approach:

Show every Real number is the limit of a sequence of rationals:
For rationals, use the constant sequence; for irrationals x, use the decimal
approximation of x, and cut it at the n-th spot (and apend 0's to the right), i.e.

x=ao.a1a2...am... --> x':=ao.a1a2...am00000...0...

Then x' is rational, and |x-x'|< 10^{-m}

For any accuracy you want, adjust m, i.e., let it go as far as you want.